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We consider a recent formulation of weak KAM theory proposed by Evans. Aswell as for classical integrability, for one dimensional mechanical Hamiltoniansystems all the computations can be explicitly done. This allows us on the onehand to illustrate the geometric content of the theory, on the other hand toprove new lower bounds which extend also to the generic n degrees of freedomcase.

A new estimate on Evans' Weak KAM approach